Page 154 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 5.3 Coordinate Coupling 141

Figure 5.3-5.

The equations of motion indicate static coupling.
mx + /cj(jc - /,0) + k2(x + I2O) = 0
J^.6 — k^(x - / | ^ ) / i -t- k2(x + /2 ^ )^2 ^ ^
Assuming harmonic motion, we have

(A:j k2 - ~
- ^2^2) + ^2^1 - ^^1)
From the determinant of the matrix equation, the two natural frequencies are
wj = 6.90rad/s == 1.10 cps

(02 = 9.06 rad/s = 1.44 cps
The amplitude ratios for the two frequencies are

= -1 4 .6 ft/ra d = -3.06 in./deg

{l]<o ^ ^ in./deg

The mode shapes are illustrated by the diagrams of Fig. 5.3-6.
In interpreting these results, the first mode, o>| =6.9 rad/s is largely vertical
translation with very small rotation, whereas the second mode, 0)2 = 9.06 rad/s is mostly

Figure 5.3-6. Normal modes of the
system shown in Figure 5.3-5.